Foundations Reference
Equations of Lines
A line is determined by its slope and one point, or by two points. The common task is to insert the known point and slope into $y=mx+b$, solve for $b$, then write the finished equation.
Fact Table
| Given | Best Form | What to Do |
|---|---|---|
| Slope and y-intercept. | $y=mx+b$ | Insert $m$ and $b$ directly. |
| Slope and one point. | $y=mx+b$ or point-slope form. | Insert the point for $(x,y)$ and solve for $b$. |
| Two points. | Start with the slope formula. | Find $m$, then use either point to solve for $b$. |
| Horizontal line. | $y=c$ | The slope is $0$. |
| Vertical line. | $x=c$ | The slope is undefined, so it is not $y=mx+b$. |
Content Formulas
Slope-Intercept Form
$$y=mx+b$$
Point-Slope Form
$$y-y_1=m(x-x_1)$$
Slope Formula
$$m=\frac{y_2-y_1}{x_2-x_1}$$
Parallel and Perpendicular Slopes
$$m_{\parallel}=m$$ $$m_{\perp}=-\frac1m$$
A point like $(4,-3)$ means $x=4$ and $y=-3$. When using $y=mx+b$, substitute both coordinates and the slope before solving for $b$.
Classic Examples
Point and Slope
Find an equation of the line that passes through $(4,-3)$ and has slope $2$.
Line MoveUse the point for $(x,y)$ and the slope for $m$ in $y=mx+b$, then solve for $b$.
$$y=mx+b$$
$$-3=2(4)+b$$
$$-3=8+b$$
$$b=-11$$
$$y=2x-11$$
Point-Slope First
Write the line through $(-2,5)$ with slope $-\frac32$.
Line MovePoint-slope form keeps the given point visible. Simplify only after the substitution is clear.
$$y-y_1=m(x-x_1)$$
$$y-5=-\frac32(x-(-2))$$
$$y-5=-\frac32(x+2)$$
$$y=-\frac32x-3+5$$
$$y=-\frac32x+2$$
Two Points
Find the equation of the line through $(1,4)$ and $(5,12)$.
Line MoveFind the slope first, then use one point to solve for $b$.
$$m=\frac{12-4}{5-1}$$
$$m=2$$
$$4=2(1)+b$$
$$b=2$$
$$y=2x+2$$
Parallel and Perpendicular Lines
A line has slope $-\frac23$. Find the slope of a parallel line and a perpendicular line.
Line MoveParallel slopes match. Perpendicular slopes are opposite reciprocals.
$$m_{\parallel}=-\frac23$$
$$m_{\perp}=\frac32$$